Theorem toponcom 22950

Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
toponcom	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Proof of Theorem toponcom

Step	Hyp	Ref	Expression
1		toponuni 22936	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐽 = ∪ 𝐾)
2	1	eqcomd 2741	. . 3 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐾 = ∪ 𝐽)
3	2	anim2i 617	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
4		istopon 22934	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
5	3, 4	sylibr 234	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∪ cuni 4912  ‘cfv 6563  Topctop 22915  TopOnctopon 22932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-topon 22933

This theorem is referenced by:  toponcomb  22951  kgencn3  23582